# §35.6 Confluent Hypergeometric Functions of Matrix Argument

## §35.6(i) Definitions

 35.6.1 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}{k!% }\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}}{{\left[b\right]_{\kappa}}}Z% _{\kappa}\left(\mathbf{T}\right).$
 35.6.2 $\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}+\mathbf{X}\right|}^{% b-a-\frac{1}{2}(m+1)}\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.

### Laguerre Form

 35.6.3 $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+\nu+% \frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*{{}_% {1}F_{1}}\left({-\nu\atop\gamma+\frac{1}{2}(m+1)};\mathbf{T}\right),$ $\Re\left(\gamma\right),\Re\left(\gamma+\nu\right)>-1$. ⓘ Defines: $L^{(\NVar{\gamma})}_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$: Laguerre function of matrix argument Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$: generalized hypergeometric function of matrix argument, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$: multivariate gamma function, $\Re$: real part, $\mathbf{T}$: real symmetric $m\times m$ matrix and $m$: positive integer Permalink: http://dlmf.nist.gov/35.6.E3 Encodings: TeX, pMML, png See also: Annotations for §35.6(i), §35.6(i), §35.6 and Ch.35

## §35.6(ii) Properties

 35.6.4 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left(a% ,b-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\mathrm{etr}% \left(\mathbf{T}\mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}% \left|\mathbf{I}-\mathbf{X}\right|^{b-a-\frac{1}{2}(m+1)}\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b-a\right)>\frac{1}{2}(m-1)$.
 35.6.5 $\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{b-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a\atop b};\mathbf{S}% \mathbf{X}\right)\mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(b\right)\left|\mathbf{% I}-\mathbf{S}\mathbf{T}^{-1}\right|^{-a}\left|\mathbf{T}\right|^{-b},$ $\mathbf{T}>\mathbf{S}$, $\Re\left(b\right)>\frac{1}{2}(m-1)$.
 35.6.6 $\mathrm{B}_{m}\left(b_{1},b_{2}\right)\left|\mathbf{T}\right|^{b_{1}+b_{2}-% \frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}% \right)=\int_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}\left|\mathbf{X}\right|^{% b_{1}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}\atop b_{1}};\mathbf{X}\right)% {\left|\mathbf{T}-\mathbf{X}\right|}^{b_{2}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}% \left({a_{2}\atop b_{2}};\mathbf{T}-\mathbf{X}\right)\mathrm{d}{\mathbf{X}},$ $\Re\left(b_{1}\right),\Re\left(b_{2}\right)>\frac{1}{2}(m-1)$.
 35.6.7 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\mathrm{etr}\left(\mathbf{T}% \right){{}_{1}F_{1}}\left({b-a\atop b};-\mathbf{T}\right).$
 35.6.8 $\int_{\boldsymbol{\Omega}}\left|\mathbf{T}\right|^{c-\frac{1}{2}(m+1)}\Psi% \left(a;b;\mathbf{T}\right)\mathrm{d}{\mathbf{T}}=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(a-c\right)\Gamma_{m}\left(c-b+\frac{1}{2}(m+1)\right)}{% \Gamma_{m}\left(a\right)\Gamma_{m}\left(a-b+\frac{1}{2}(m+1)\right)},$ $\Re\left(a\right)>\Re\left(c\right)+\frac{1}{2}(m-1)>m-1$, $\Re\left(c-b\right)>-1$.

## §35.6(iii) Relations to Bessel Functions of Matrix Argument

 35.6.9 $\lim_{a\to\infty}{{}_{1}F_{1}}\left({a\atop\nu+\frac{1}{2}(m+1)};-a^{-1}% \mathbf{T}\right)=\frac{A_{\nu}\left(\mathbf{T}\right)}{A_{\nu}\left(% \boldsymbol{{0}}\right)}.$
 35.6.10 $\lim_{a\to\infty}\Gamma_{m}\left(a\right)\Psi\left(a+\nu;\nu+\tfrac{1}{2}(m+1)% ;a^{-1}\mathbf{T}\right)=B_{\nu}\left(\mathbf{T}\right).$

## §35.6(iv) Asymptotic Approximations

For asymptotic approximations for confluent hypergeometric functions of matrix argument, see Herz (1955) and Butler and Wood (2002).